This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Theorem mpbi2and

Description: Detach a conjunction of truths in a biconditional. (Contributed by NM, 6-Nov-2011) (Proof shortened by Wolf Lammen, 24-Nov-2012)

Ref Expression
Hypotheses mpbi2and.1 ⊢ φ → ψ
mpbi2and.2 ⊢ φ → χ
mpbi2and.3 ⊢ φ → ψ ∧ χ ↔ θ
Assertion mpbi2and ⊢ φ → θ

Proof

Step Hyp Ref Expression
1 mpbi2and.1 ⊢ φ → ψ
2 mpbi2and.2 ⊢ φ → χ
3 mpbi2and.3 ⊢ φ → ψ ∧ χ ↔ θ
4 1 2 jca ⊢ φ → ψ ∧ χ
5 4 3 mpbid ⊢ φ → θ